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A Characterization of Strong Completeness in Fuzzy Metric Spaces

Author

Listed:
  • Valentín Gregori

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, Spain)

  • Juan-José Miñana

    (Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5, 07122 Palma, Spain)

  • Bernardino Roig

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, Spain)

  • Almanzor Sapena

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/ Paranimf, 1, 46730 Grao de Gandia, Spain)

Abstract

Here, we deal with the concept of fuzzy metric space ( X , M , ∗ ) , due to George and Veeramani. Based on the fuzzy diameter for a subset of X , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.

Suggested Citation

  • Valentín Gregori & Juan-José Miñana & Bernardino Roig & Almanzor Sapena, 2020. "A Characterization of Strong Completeness in Fuzzy Metric Spaces," Mathematics, MDPI, vol. 8(6), pages 1-11, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:861-:d:363033
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    References listed on IDEAS

    as
    1. Gregori, V. & Romaguera, S. & Veeramani, P., 2006. "A note on intuitionistic fuzzy metric spaces," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 902-905.
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